Search

It is well-known that random attractors of a random dynamical system are generally not unique. We show that for general pullback attractors and weak attractors, there is always a minimal (in the sense of smallest) random attractor which attracts a given family of (possibly random) sets. We provide an example which shows that this property need not hold for forward attractors. We point out that our concept of a random attractor is very general: The family of sets which are attracted is allowed to be completely arbitrary. [ABSTRACT FROM AUTHOR]

It is well-known that a stochastic differential equation (sde) on a Euclidean space driven by a (possibly infinite-dimensional) Brownian motion with Lipschitz coefficients generates a stochastic flow of homeomorphisms. If the Lipschitz condition is replaced by an appropriate one-sided Lipschitz condition (sometimes called monotonicity condition) and the number of driving Brownian motions is finite, then existence and uniqueness of global solutions for each fixed initial condition is also well-known. In this paper we show that under a slightly stronger one-sided Lipschitz condition the solutions still generate a stochastic semiflow which is jointly continuous in all variables (but which is generally neither one-to-one nor onto). We also address the question of strong Δ-completeness which means that there exists a modification of the solution which if restricted to any set A ⊂ R d of dimension Δ is almost surely continuous with respect to the initial condition. [ABSTRACT FROM AUTHOR]

Beck et al. (2013) introduced a new distribution class J which contains many heavy-tailed and light-tailed distributions obeying the principle of a single big jump. Using a simple transformation which maps heavy-tailed distributions to light-tailed ones, we find some light-tailed distributions, which belong to the class J but do not belong to the convolution equivalent distribution class and which are not even weakly tail equivalent to any convolution equivalent distribution. This fact helps to understand the structure of the light-tailed distributions in the class J and leads to a negative answer to an open question raised by the above paper. [ABSTRACT FROM AUTHOR]

Abstract: There is a long-standing conjecture of Nussbaum which asserts that every finite set in on which a cyclic group of sup-norm isometries acts transitively contains at most points. The existing evidence supporting Nussbaum’s conjecture only uses abelian properties of the group. It has therefore been suggested that Nussbaum’s conjecture might hold more generally for abelian groups of sup-norm isometries. This paper provides evidence supporting this stronger conjecture. Among other results, we show that if is an abelian group of sup-norm isometries that acts transitively on a finite set in and contains no anticlockwise additive chains, then has at most points. [Copyright &y& Elsevier]

We state and prove a Local Stable Manifold Theorem (Theorem 4.1) for non-linear stochastic differential systems with finite memory (viz. stochastic functional differential equations (sfde''s)). We introduce the notion of hyperbolicity for stationary trajectories of sfde''s. We then establish the existence of smooth stable and unstable manifolds in a neighborhood of a hyperbolic stationary trajectory. The stable and unstable manifolds are stationary and asymptotically invariant under the stochastic semiflow. The proof uses infinite-dimensional multiplicative ergodic theory techniques developed by D. Ruelle, together with interpolation arguments. [Copyright &y& Elsevier]

We consider non-linear stochastic functional differential equations (sfde''s) on Euclidean space. We give sufficient conditions for the sfde to admit locally compact smooth cocycles on the underlying infinite-dimensional state space. Our construction is based on the theory of finite-dimensional stochastic flows and a non-linear variational technique. In Part II of this article, the above result will be used to prove a stable manifold theorem for non-linear sfde''s. [Copyright &y& Elsevier]

We study stochastic differential equations (SDEs) with multiplicative Stratonovich-type noise of the form d X t = b (X t) d t + σ (X t) ∘ d W t , X 0 = x 0 ∈ R d , t ⩾ 0 , with a possibly singular drift b ∈ L p (R d) , p > d and p ⩾ 2 , and show that such SDEs can be approximated by random ordinary differential equations by smoothing the noise and the singular drift at the same time. We further prove a support theorem for this class of SDEs in a rather simple way using the Girsanov theorem. [ABSTRACT FROM AUTHOR]

Abstract: We study the quantization problem for certain types of jump processes. The probabilities for the number of jumps are assumed to be bounded by Poisson weights. Otherwise, jump positions and increments can be rather generally distributed and correlated. We show in particular that in many cases entropy coding error and quantization error have distinct rates. Finally, we investigate the quantization problem for the special case of -valued compound Poisson processes. [Copyright &y& Elsevier]